# Some Properties of the Fuss–Catalan Numbers

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction and Main Results

**Theorem**

**1.**

**Theorem**

**2.**

- 1.
- the sequence ${\left\{{\mathcal{A}}_{n}(p,r)\right\}}_{n\ge 0}$,$${\mathcal{A}}_{n}(p,r)=\left\{\begin{array}{cc}1,\hfill & n=0,\hfill \\ {\displaystyle \frac{1}{\sqrt[n\phantom{\rule{-0.166667em}{0ex}}]{{A}_{n}(p,r)}}},\hfill & n\in \mathbb{N},\hfill \end{array}\right.$$
- 2.
- the sequence of the Fuss–Catalan numbers ${\left\{{A}_{n}(p,r)\right\}}_{n\ge 0}$ is increasing and logarithmically convex.

**Theorem**

**3.**

## 2. Lemmas

**Lemma**

**1**

**Lemma**

**2**

- 1.
- the unique zero ${x}_{0}$ of the equation$$\frac{\psi (x+b)-\psi (x+a)}{lnb-lna}=1$$
- 2.
- when $b>a$, the function $C(a,b;x)$ is decreasing in $x\in [0,{x}_{0})$, increasing in $x\in ({x}_{0},\infty )$, and logarithmically convex in $x\in [0,\infty )$;
- 3.
- when $b<a$, the function $C(a,b;x)$ is increasing in $x\in [0,{x}_{0})$, decreasing in $x\in ({x}_{0},\infty )$, and logarithmically concave in $x\in [0,\infty )$.

## 3. Proofs of Theorems 1–3

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**3.**

## 4. Remarks

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Qi, F.; Cerone, P. Some Properties of the Fuss–Catalan Numbers. *Mathematics* **2018**, *6*, 277.
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Qi F, Cerone P. Some Properties of the Fuss–Catalan Numbers. *Mathematics*. 2018; 6(12):277.
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Qi, Feng, and Pietro Cerone. 2018. "Some Properties of the Fuss–Catalan Numbers" *Mathematics* 6, no. 12: 277.
https://doi.org/10.3390/math6120277